Upper Semicontinuity Of Random Attracors For The Generalized Ginzburg-Landau Equation With Multiplicative Noise

In this paper,we study the asymptotic behavior of solutions for one dimension generalized Ginzburg-Landau equation with multiplicative noise and prove that the random dynamical system generated by the unique solution of corresponding prob-lem exists a random attractor in L2 space,which is upper semi-continuous.We will investigate the following generalized Ginzburg-Landau equation with multiplicative noise: du+(λux-γuxx-u)dt=(β|u|2u-δ|u|4u-μu2-ux-v|u|2ux)dt+εu(?)dW(t), where ε>0,β=β1+iβ2,γ=γ1+γ2,δ=δ1+iδ2,μ=μ1+iμ2,v=v1+iv2 are complex and k,βi,γi,δi,μi,vi(i=1,2)are real numbers.We assume that γ1> 0,δ1>0,and γ1δ1>|μ|2+|v|2.The white noise described by a process W(t),which is a Wiener process on the s-tandard metric dynamical system (Ω,F,P,θt),where Ω={ω∈C(R,R):ω(0)=0}. F is the Borel σ-algebra induced by the compact-open topology of Ω,P is a Wiener measure,for ω∈Ω and t∈R,θ satisfies θtω(·)=ω(·+t)-ω(t).This equation is supplemented with the boundary and initial conditions as follows: u(0,t)=u(1,t)=0, t≥0, u(x,0)=u0(x), x∈[0,1].This paper is divided into four chapters:In the first chapter, we introduce the background on the theory of random dynamical system, random attractors and the research status of Ginzburg-Landau equation, main contents of this paper, as well as some preliminary definitions and results, which will be used.In the second chapter, there is no stochastic differential for introducing O-U transformation. The unique solution of the equation by using the Galerkin approx-imate method is obtained, which generates a continuous random dynamical system.In the third chapter, we get the unique random attractor in L2(0,1) space by uniform estimates of solutions and proving that there exist random absorbing sets both in L2(0,1) space and H01(0,1) space, which combines with Sobolev compact embedding theory.In the fourth chapter, by the convergence of random dynamical system in L2 space, upper semicontinuity of random attractor is obtained.